In mathematics, the Poincaré lemma gives a sufficient condition for a closed differential form to be exact (while an exact form is necessarily closed). Precisely, it states that every closed p-form on an open ball in Rn is exact for p with 1 ≤ p ≤ n. The lemma was introduced by Henri Poincaré in 1886.
Informal discussion Especially in calculus, the Poincaré lemma also says that every closed 1-form on a simply connected open subset in R n {\displaystyle \mathbb {R} ^{n}} is exact. In simpler terms, it means that if a differential form is closed in a region that can be shrunk to a point, then it can be written as the derivative of another form; i.e. if dα = 0 on a simply connected region, we can always find α = dβ; therefore we have d(dβ) = 0, expressed simply as d2 = 0. This concept is used in mathematical physics, particularly in the context of electromagnetism and differential geometry, where it relates to the fact that the boundary of a boundary is always empty, i.e. if you have a surface (a 2-form) and you take its boundary (a 1-form, a curve), then the boundary of that boundary (a 0-form, a point) is an empty set. In electromagnetism, magnetic fields can be described using a vector potential, and the Poincaré lemma helps in finding such potentials when the magnetic field is "well-behaved" (i.e., when the magnetic field is not due to a monopole), Gauss's law for magnetism states that the total magnetic flux through a closed surface is always zero, which implies that magnetic monopoles, if they exist, are not isolated but must be accompanied by other magnetic charges. In the language of cohomology, the Poincaré lemma says that the k-th de Rham cohomology group of a contractible open subset of a manifold M (e.g., M = R n {\displaystyle M=\mathbb {R} ^{n}} ) vanishes for k ≥ 1 {\displaystyle k\geq 1} . In particular, it implies that the de Rham complex yields a resolution of the constant sheaf R M {\displaystyle \mathbb {R} _{M}} on M. The singular cohomology of a contractible space vanishes in positive degree, but the Poincaré lemma does not follow from this, since the fact that the singular cohomology of a manifold can be computed as the de Rham cohomology of it, that is, the de Rham theorem, relies on the Poincaré lemma. It does, however, mean that it is enough to prove the Poincaré lemma for open balls; the version for contractible manifolds then follows from the topological consideration. The Poincaré lemma is also a special case of the homotopy invariance of de Rham cohomology; in fact, it is common to establish the lemma by showing the homotopy invariance or at least a version of it.
Proofs A standard proof of the Poincaré lemma uses the homotopy invariance formula (cf. see the proofs below as well as Integration along fibers#Example). The local form of the homotopy operator is described in Edelen (2005) and the connection of the lemma with the Maurer–Cartan form is explained in Sharpe (1997).
Direct proof The Poincaré lemma can be proved by means of integration along fibers. (This approach is a straightforward generalization of constructing a primitive function by means of integration in calculus.) We shall prove the lemma for an open subset U ⊂ R n {\displaystyle U\subset \mathbb {R} ^{n}} that is star-shaped or a cone over [ 0 , 1 ] {\displaystyle [0,1]} ; i.e., if x {\displaystyle x} is in U {\displaystyle U} , then t x {\displaystyle tx} is in U {\displaystyle U} for 0 ≤ t ≤ 1 {\displaystyle 0\leq t\leq 1} . This case in particular covers the open ball case, since an open ball can be assumed to be centered at the origin without loss of generality. The trick is to consider differential forms on U × [ 0 , 1 ] ⊂ R n + 1 {\displaystyle U\times [0,1]\subset \mathbb {R} ^{n+1}} (we use t {\displaystyle t} for the coordinate on [ 0 , 1 ] {\displaystyle [0,1]} ). First define the operator π ∗ {\displaystyle \pi _{*}} (called the fiber integration) for k-forms on U × [ 0 , 1 ] {\displaystyle U\times [0,1]} by
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